Abstract We propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with gradient flow structure. These properties allow accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features the are essential. The proposed is able to cope non-smooth states, different time scales including metastability, as well concentrations self-similar behavior induced singular kernels. use...

Abstract Aquatic bacteria like Bacillus subtilis are heavier than water yet they able to swim up an oxygen gradient and concentrate in a layer below the surface, which will undergo Rayleigh–Taylor-type instabilities for sufficiently high concentrations. In literature, simplified chemotaxis–fluid system has been proposed as model bio-convection modestly diluted cell suspensions. It couples convective chemotaxis oxygen-consuming oxytactic with incompressible Navier–Stokes equations subject...

We propose a PDE chemotaxis model, which can be viewed as regularization of the Patlak-Keller-Segel (PKS)system. Our modification is based on fundamental physical property chemotactic flux function---itsboundedness. This means that cell velocity proportional to magnitude chemoattractant gradientonly when latter small, while gradient tends infinity velocitysaturates. Unlike original PKS system, solutions modified model do not blow up in either finiteor infinite time any number spatial...

Summary Shallow water models are widely used to describe and study free‐surface flow. While in some practical applications the bottom friction does not have much influence on solutions, there still many applications, where is important. In particular, terms will play a significant role when depth of very small. this paper, we shallow equations with develop semi‐discrete second‐order central‐upwind scheme that capable exactly preserving physically relevant steady states maintaining positivity...

In this paper, we develop a family of second-order semi-implicit time integration methods for systems ordinary differential equations (ODEs) with stiff damping term. The important feature the new resides in fact that they are capable exactly preserving steady states as well maintaining sign computed solution under step restriction determined by nonstiff part system only. based on modification explicit strong stability Runge--Kutta (SSP-RK) and proven to have formal second order accuracy,...

In this talk, we consider a mathematical model of cloud physics that consists the Navier-Stokes equations coupled with evolution for water vapor, water, and rain. model, describe weakly compressible flows viscous heat conductivity effects, while microscale is modeled by system advection-diffusion-reaction equations. We aim to explicitly uncertainties arising from unknown input data, such as parameters initial or boundary conditions. The developed stochastic Galerkin method combines...

The purpose of this paper is to provide global existence and uniqueness results for a family fluid transport equations by establishing convergence the particle method applied these equations. considered PDEs collection strongly nonlinear which yield traveling wave solutions can be used model variety flows in dynamics. We apply studied evolutionary new self-contained proving its convergence. latter accomplished using concept space-time bounded variation associated compactness properties. From...

We first present a new sticky particle method for the system of pressureless gas dynamics. The is based on idea particles, which seems to work perfectly well models with point mass concentrations and strong singularity formations. In this method, solution sought in form linear combination $\delta$-functions, whose positions coefficients represent locations, masses, momenta respectively. locations particles are then evolved time according ODEs, obtained from weak formulation PDEs. velocities...

In this paper, we introduce and study one-dimensional models for the behavior of pedestrians in a narrow street or corridor. We begin at microscopic level by formulating stochastic cellular automata model with explicit rules moving two opposite directions. Coarse-grained mesoscopic macroscopic analogs are derived leading to coupled system PDEs density pedestrian traffic. The obtained first-order conservation laws is only conditionally hyperbolic. also derive higher-order nonlinear diffusive...

The equation ∂ t u = u∂ xx 2 − ( c 1)(∂ x ) is known in literature as a qualitative mathematical model of some biological phenomena. Here this derived the groundwater flow water-absorbing fissurized porous rock; therefore, we refer to filtration-absorption equation. A family self-similar solutions constructed. Numerical investigation evolution non-self-similar Cauchy problems having compactly supported initial conditions performed. experiments indicate that obtained represent intermediate...

Abstract Systems of convection–diffusion equations model a variety physical phenomena which often occur in real life. Computing the solutions these systems, especially convection dominated case, is an important and challenging problem that requires development fast, reliable accurate numerical methods. In this paper, we propose second‐order fast explicit operator splitting (FEOS) method based on Strang splitting. The main idea to solve parabolic via discretization formula for exact solution...

This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and separated by material interfaces, each endowed its own equation of state (EOS). Cell averages computational cells that occupied several require a "mixed-cell" EOS, which may not always be physically meaningful, often leads to spurious oscillations. We present new interface tracking algorithm, avoids using mixed-cell information solving the Riemann problem...